Finite element methods for the surface Stokes problem
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Title: Finite element methods for the surface Stokes problem
Abstract: Surface Stokes equations have attracted significant recent attention in numerical analysis because approximation of their solutions poses significant obstacles not encountered in the Euclidean context. One of these challenges involves simultaneously enforcing tangentiality and continuity of discrete velocity approximations. Existing finite element methods all enforce one of these two constraints weakly either by penalization or by use of Lagrange multipliers. However, a robust and systematic construction of surface Stokes finite element spaces with nodal degrees of freedom is still missing. In this talk, we introduce a novel approach addressing these challenges by constructing surface finite element spaces with tangential velocity fields. Functions in the discrete spaces are not continuous, but do have conormal-continuity, and the resulting methods do not require ad hoc penalization. We prove stability and optimal-order energy-norm convergence of the method and provide numerical examples illustrating the theory. This is joint work with Alan Demlow, Texas A&M.